A terahertz meta-sensor array for 2D strain mapping

Large-scale stretchable strain sensor arrays capable of mapping two-dimensional strain distributions have gained interest for applications as wearable devices and relating to the Internet of Things. However, existing strain sensor arrays are usually unable to achieve accurate directional recognition and experience a trade-off between high sensing resolution and large area detection. Here, based on classical Mie resonance, we report a flexible meta-sensor array that can detect the in-plane direction and magnitude of preloaded strains by referencing a dynamically transmitted terahertz (THz) signal. By building a one-to-one correspondence between the intrinsic electrical/magnetic dipole resonance frequency and the horizontal/perpendicular tension level, arbitrary strain information across the meta-sensor array is accurately detected and quantified using a THz scanning setup. Particularly, with a simple preparation process of micro template-assisted assembly, this meta-sensor array offers ultrahigh sensor density (~11.1 cm−2) and has been seamlessly extended to a record-breaking size (110 × 130 mm2), demonstrating its promise in real-life applications.


■ Supplementary Table List
Table S1.Comparison of our meta-sensor array with previous representative strain sensor arrays works mentioned in Fig. 1c of the main text.Table S2.Measured MD and ED resonance information across the 10 × 10 girds on the sample surface by 2D strain mapping.Table S3.Calculation of the strain values across the 10 × 10 girds on the sample surface by means of visual computational analysis (original length and the length after strain of the mesh for y direction).
Table S4.Calculation of the strain values across the 10 × 10 girds on the sample surface by means of visual computational analysis (original length and the length after strain of the mesh for x direction).Considering the first level electric and magnetic dipole resonance, the scattering coefficients of electric and magnetic dipoles can be expressed by 1 a , 1 b respectively as
From the Clausius-Mossotti formula, it follows that, in the case of spherical dielectric particles with dimensions much smaller than the wavelength of the incident electromagnetic wave their equivalent permittivity and equivalent permeability can be expressed as where is the volume fraction of the spherical media particles and N is the unit density of the microsphere particles.Also based on the Mie medium scattering theory, the above equations are combined to give (1) Electric dipole resonance When a1 is infinite,the subwavelength spherical particle can excite a strong electric dipole, then from the Eq. ( 5) can be If p ε ≫ h ε , that is to say, if the dielectric constant of the dielectric sphere is much larger than the dielectric constant of the surrounding medium, it can be obtained by Eq. ( 9): Substituting above Equation into Eq.( 7) the equation, we get: The first correct solution of the above formula is n=4.49, from which the minimum resonance frequency of the electric dipole can be calculated as follows 4.49 2 (2) Magnetic dipole resonance When b1 is infinite, the subwavelength spherical particles can excite strong magnetic dipoles, then it can be obtained by Eq. ( 6) Taking above equation into Eq.( 7) gives sin( ) 0, The root of Eq. ( 14) is: , 1,2,3,... nx q q π = = (15) When q=1, the frequency of the minimum resonance of the magnetic dipole is 2

Note S2. Analysis of physical mechanisms of strain-dependent THz response based on electromagnetic coupling between microspheres.
We analyze the effect of the electromagnetic near-field couplings on the Mie resonance in the ZrO2 resonator array shown in Fig. S2.Both the impacts of the applied strain and the induced strain are incorporated in the Lagrangian analysis.However, the induced strain has a weaker impact on the near-field couplings than the applied strain since the compression due to induced strain is very limited.For example, in the ED resonance, the Lagrangian of the coupled resonator system can be expressed as The coupling effects described in above equation only take the four nearest neighboring resonators, which are marked with red circles, as shown in Fig. S2 The subscripts for the coupling coefficients ED ED we arrive at Since the plane-wave excitation is uniform across the array, the coupled dielectric resonator array operates in the symmetrical mode.Thus, all the resonators have the same charge distributions, and they oscillate collectively in phase, yielding ( ) . Solving Eq. ( 19) yields the ED ransonance eigenfrequency of the coupled ZrO2 array can be obtained as where the decoupled resonance frequency is f0.By referring to the near-field expressions for the infinitesimal electric dipole, these coefficients can be linked to the unit cell sizes Px and Py as well as the wavenumber k in the PDMS via S21 To uncover the effect of the substrate PDMS on the ED and MD resonances, we introduced PDMS into the initial ZrO2 microsphere array (Fig. 2c of the main text, and Fig. S4a) for simulation that was used for establishing the basic sensing mechanism.As shown in Fig. S15b, the PDMS is gradually added towards two opposite directions from the reference plane (dot line shown in Fig. S4b) at the center of the ZrO2 sphere.When Δh increases from 0 to 260 μm, three stages can be observed from the ED and MD resonance shift.
Stage 1: the ED and MD resonances appear large shift (Fig. S4c), when the PDMS increases from 0 to wrapping the ZrO2 sphere just right (namely Δh is equal to 80 μm).During this process, the MD resonance gradually moves to higher frequency while the ED one shifts to the opposite direction, which can be attributed by the decayed ED and MD amplitude of the ZrO2 microsphere array in the PDMS substrate (Fig. S4g) compared to that in the vacuum background (Fig. S4f).
Stage 2: due to the little variation in E-field and H-field distribution (Figs.S4g and h), the ED resonance only appears fine shift (Fig. S4c) while the MD resonance is almost fixed, when the PDMS increases from 80 to 100 μm.
Stage 3: when Δh is larger than 100 μm, the ED and MD resonance frequency will keep stable (Fig. S4e) as a result of the almost same E-field and H-field distributions (Figs.S4h and i)).In addition, with the increase of Δh, more THz energy can be consumed through the dielectric loss of the PDMS substrate.So that, fewer E-field can be observed localized in the PDMS substrate (white wireframes in Figs.S4h-i).That's the reason why the transmittance at the ED resonance gradually decays as Δh increases.As shown in Fig. S5a for the fabrication process of the meta-sensor array.Initially, a porous template with a hexagonal distribution of holes is produced by lithography.Specifically, the hole diameter is 5 µm, the hole depth is 5 µm, and the holes are 10 µm apart from each other.

Supplementary
The liquid PDMS after centrifugation and defoaming was then scraped and coated on the above porous template to obtain a liquid film of about 80 μm thick.Then, it was placed in a vacuum oven at 70 ℃ for about 10 minutes.At this stage, although the liquid PDMS has been cured, the surface still retains a specific viscosity.Next, the screen-printing template is attached to the PDMS surface and microspheres are placed on it.The microspheres fall into the holes under the thrust of the soft brush and their own gravity, forming an array of microspheres.Finally, an integrated stretchable meta-sensor was obtained by encapsulating it with liquid PDMS and curing it at 70 ℃ for ~ 25 minutes in a vacuum oven.The cross-sectional view of the resulting meta-sensor is shown in Fig. S5b, which clearly shows the surface superhydrophobic layer, and the array of ZrO2 microspheres embedded in the PDMS matrix.The superhydrophobic layer is obtained by the shape of the porous template.The morphology of the sample stripped from the porous template is a column array with narrow top and wide bottom (upper right of Fig. S5b).
After flame treating the sample surface for 3~10 s, secondary nanostructures can be formed on the surface of the columnar structure (lower right of Fig. S5b).It is the result of the aggregation of the organosilicon compound in the PDMS matrix on the surface, which can enhance the superhydrophobicity of the sample surface S25,S26 .In this method described above, the processing area also dependent on the area of the employed template.By using a large-scale template, one can obtain an ultra-large-scale meta-sensor array, which is highly desirable in practical applications.

Note S5. Analysis of the minimum detectable strain variations of our device.
Considering that the simulated strain-resonance frequency curves (Figs.3f-g of the main text) are monotonically smooth and continuous, our device theoretically has the capability to detect an ultra-small strain variation.However, in practice, confined by the spectral resolution (0.001 THz) of the THz time-domain spectroscopy (TDS) device (QT-TRS1000, Quenda, China), the smallest strain variation detection performance of our device is limited.
In principle, the smallest strain variation corresponds to the point on the curves with the steepest slope.Regarding the x-directional strain (inset ⅱ of Fig. 3f of the main text), the slope reaches its maximum value when the applied strain is equal to 65%.Conversely, strainresonance frequency curve in inset ⅰ of Fig. 3g of the main text has the greatest slop with the externally y-directional strain of 0%.In the meantime, considering the spectral resolution of the TDS system (0.001 THz), the theoretically smallest strain variation value (0.63% @ x direction and 2.7% @ y direction) can be successfully obtained through shifting the resonance frequency by 0.001 THz towards the lower slop direction as shown in the insets of Figs.3f-g.Then the corresponding test under the strain condition obtained above was conducted to observe the resonance frequency shifting.From Figs. 3i-j of the main text, ~0.001 THz resonance frequency shifting can be implemented in the experiment, which is consistent with the simulated predictions (inset ⅱ of Fig. 3f and inset ⅰ of Fig. 3g of the main text).We statistically analyze the above measured results.Trough 100-round test (Fig. S8), the average resonance frequency and the standard deviation can be calculated as [0.3511THz/0.0002THz @ 64.37% strain along x direction; 0.3500 THz/0.0004THz @ 65% strain along x direction] (Fig. S8a) and [0.3221 THz/0.0004THz @ 0 % strain along y direction; 0.3210 THz/0.0005THz @ 2.7% strain along y direction] (Fig. S8b).Therefore, the smallest strain variation that can be detected by our design was finalized as 0.63% @ x direction and 2.7% @ y direction.Their difference can be attributed to the discrepancy of the corresponding resonance frequency shift bandwidth.

Note S6. Calculation of the GF of our device.
By analogy, we define the ratio of the relative frequency shift of the resonance peak to the strain as the GF of our device as follows where ∆f is the resonance frequency shift value from the initial resonance frequency f0, and ∆ε denotes the applied strain.Then, by substituting the measured results (Figs. 3f-g in the main text) into Eq.( 24), strain-Δf/f0 curves and the corresponding fitted ones can be obtained as shown in Fig. 3k of the main text.Following, the maximum slopes of the fitted curves, which correspond to the maximum GF, are calculated to be ~0.413@ x direction and ~0.09 @ y direction.

Table S2. Measured MD and ED resonance information across the 10 × 10 girds on the sample surface by 2D strain mapping.
Position @MD @ED Position @MD @ED Position @MD @ED Position @MD @ED ( Table S3.Calculation of the strain values across the 10 × 10 girds on the sample surface by means of visual computational analysis (original length and the length after strain of the mesh for y direction).Note S7.Self-cleaning effect of the meta-sensor array.
For practical applications, it is essential to keep the sensor surface away from dusts, water and other interferents with strong THz adsorption.Thus self-cleaning capacity mimicing the superhydrophobic lotus leaf (Fig. S12a) was introdued to the meta-sensor array with a compatible fabrication process.Using replica method, micro pillar array was fabricated on the surface and nano organosilicon compound was further generated on the surface of the micro pillar array after flame treating for 3~10 s (Fig. S12b).Static contact angle (CA) and roll-off angle were measured from sessile water drops with a drop shape analysis instrument (DSA 100, Kruss, Hamburg) at room temperature.With the hierarchical structures and low surface energy of PDMS, the surface of the meta-sensor exhibit superhydrophobic with higher contact angles (CAs, ~163°) and smaller sliding angle (SA, ~4.5°) compared with the normal samples (Snormal) peeled from smooth PET (Fig. S12d, Video S1).Thus, water droplets bounce off the surface of the meta-sensor and take away the dust (with sands as a demonstration) (Figs.S12e-f, Video S2-3).These results verified the superhydrophobic and self-cleaning capacity of the fabricated meta-sensor array.
Then, the THz transmission feature of the sample with and without superhydrophobic treatment is measured considering the interference of the water and dust (Figs.S12g-h).As shown in Fig. S12g, the water covering sample (left side of Fig. S12g) absolutely loss function due to the strong THz absorption of water, while water is difficult to retain on the surface of the hydrophobically treated sample (right side of Fig. S12g)), leading to almost no influence on the transmission spectra.In contrast, the dust only decays the intensity of the resonances (left side of Fig. S12f)) and can be wiped away from the hydrophobically treated sample (right side of Fig. S12f) through flushing with the water, to keep the dynamic range of the meta-sensor.In addition, it is worth noting that the abovementioned superhydrophobic effect can be completely maintained as the strain applied to the sample gradually increases to more than 30%, denoting our design taking robust working condition adaptability (the corresponding phenomena can be found in video S4).

Note S8. Design and performance evaluation of a reflection-type meta-sensor for external strain detection.
To further verify the feasibility of the reflection-type design shown in Figs.S13a-b, we have conducted the stretching response experiments.First of all, the reflection time-domain signals of the sample applied with the unidirectional tensile strain were measured and presented in Fig. S13c (for x-directional strain) and S13d (for y-directional strain).Then, the corresponding spectra information can be obtained (Figs.S13e and h) through applying the Fourier transform In addition, bidirectional tensile strain detection performance of the proposed reflective metasensor has been evaluated.As shown in Fig. S14a, with the x-directional strain varying from 5% to 30% by 5% per step, the strain applied in the y-direction is fixed at 5%, 10%, 15%, 20%, 25% and 30%.In such cases, the ED resonance gradually shifts to a lower frequency as the xdirectional strain increases, while the MD resonance always remains localized at a certain value only associated with the y-directional deformation ratio (5% @ 0.3212 THz, 10% @ 0.3183 THz, 15% @ 0.3165 THz, 20% @ 0.3143 THz, 25% @ 0.3126 THz, and 30% @ 0.3116 THz).
By switching the strain loading, the opposite phenomenon can be revealed in Fig. S14b, i.e., the MD resonance frequency decreases with increasing strain in the y-direction, while the ED resonance always stays at a specific value, which is only related to the x-directional deformation ratio (5% @ 0.4173 THz, 10% @ 0.4148 THz, 15% @ 0.4107 THz, 20% @ 0.4072 THz, 25% @ 0.4018 THz, and 30% @ 0.3970 THz).These results demonstrate that the reflective metasensor also favours independent and noninterfering monitoring of the orthogonal strains.
Additionally, the experimental results agree well with the simulated results shown in Figs.
S15a-b, further proving the veracity of our strategy.Furthermore, in Figs.S14c-d the sensing mechanism we proposed can be directly extended to the design of reflective strain sensor elements, making it more suitable for practical applications.

Fig. S1 .
Fig. S1.Simulation results of ED and MD resonances versus microsphere spacing (Px and Py) in terms of ideal vacuum conditions.

Fig. S2 .Fig. S3 .
Fig. S2.Illustration of near-field coupling among four neighboring ZrO2-microsphere resonators.Fig. S3.Measured transmission spectra of a pure PDMS with the identical thickness to our device under different external strains.

Fig. S4 .
Fig. S4.Effect of the PDMS on ED and MD resonances.

Fig. S8 .
Fig.S8.Statistical analysis for the minimum strain variations that can be detected by our device.

Fig. S9 .
Fig. S9.Simulated transmittance spectra of the meta-sensor array simultaneously applied with bidirectional tensile strain.

Fig. S10 .
Fig. S10.Measured THz transmission spectra of the sample stretched along different angles.

Fig. S12 .
Fig. S12.Characterization of the meta-sensor array samples with self-cleaning capacity.

Fig. S15 .
Fig. S15.Simulated reflection spectra of the reflective meta-sensor array simultaneously applied with x-and y-directional tensile strain.

Fig. S16 .
Fig. S16.Measured resonance-strain response at the different locations of the fabricated sample.

Note S1 .
Mie resonance theory.Note S2.Analysis of physical mechanisms of strain-dependent THz response based on electromagnetic coupling between microspheres.Note S3.The effect of PDMS on ED and MD resonances.Note S4.Fabrication of the flexible strain meta-sensor array with self-cleaning capability.Note S5.Analysis of the minimum detectable strain variations of our device.Note S6.Calculation of the GF of our device.Note S7.Self-cleaning effect of the meta-sensor array.Note S8.Design and performance evaluation of a reflection-type meta-sensor for external strain detection.Note S9.Bending deformation response of the meta-sensor array.Note S10.Twisting deformation response of the meta-sensor array.
the magnetic (h) and electric (e) fields.Subsequently, by solving the Euler-Lagrange equationsS24

e
represent the effects of an oscillating current source radiating as an electric dipole in the near-field along transverse and longitudinal directions.While in the electric dipole coupling species, the magnetic field effect is very weak ( magnetic dipole coupling, the electric field effect is very weak ( The effect of PDMS on ED and MD resonances.

Fig. S4 .
Effect of the PDMS on ED and MD resonances.(a) Schematic diagram of the ZrO2 microsphere array in the vacuum.(b) Schematic for analyzing the effect on the ED and MD resonances.(c-e) Simulated transmittance spectra of the meta-sensor in terms of different Δh.Simulated E-field and H-field distributions across the center of the microspheres: (f) for Δh = 0 μm; (g) for Δh = 80 μm; (h) for Δh = 100 μm; (i) for Δh = 260 μm.Note S4.Fabrication of the flexible strain meta-sensor array with self-cleaning capability.
to the time signals (insets in Figs.S13c-d).Similar to the transmission-type meta-sensor, when the sample is stretched along the x-direction, the measured ED resonance gradually shifts from 0.4210 THz to 0.3640 THz as the strain increases from 0 to 65% (Figs.S13e), which is consistent with the simulation results (Figs.S13f).Conversely, when the strain along the ydirection increases from 0 to 70%, the measured MD resonance (Figs.S13h) matches the simulated one (Figs.S13i), and decreases from 0.3225 THz to 0.3000 THz.Additionally, in these two processes, the remaining resonances (the MD resonance during x-direction stretching and the ED resonance during y-direction stretching) show barely shifts (Figs.S13g and j).These experimental results have successfully proved the unidirectional tensile strain detection ability of the proposed reflective meta-sensor.
, the dynamic ED and MD resonance frequencies during stretching have been extracted from Figs.S14a-bto directly exhibit the corresponding relation between the bidirectional strains applied to our sensor and its resonance shifting.All the experimental results in Figs.S13 and S14 indicate that

work Note S1. Mie resonance theory.
The generation of resonances can be explained and predicted by Mie theoryS21-S23.A nonmagnetic dielectric sphere of radius r and refractive index n , p